Combinations of samples problem

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In summary, there are 658,008 different samples that can be chosen from a production run of 40 boards, 222,111 of which will contain at least one defective board. The probability of randomly choosing a sample of five boards that contains at least one defective board is approximately 33.76%.
  • #1
jonroberts74
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Homework Statement



Suppose that three computer boards in a production run of forty are defective. A sample of five is to be selected to be checked for defects.
a. How many different samples can be chosen?
b. How many samples will contain at least one defective board?
c. What is the probability that a randomly chosen sample of five contains at least one defective board?

The Attempt at a Solution



A) ##\binom{40}{5} = \frac{40!}{5!(35!)} = 658\;008##

B) ##\binom{5}{1}+\binom{5}{2}+\binom{5}{3} = 25##

C) ##P(E) = \frac{N(E)}{N(S)} = \frac{25}{658\;008}##
 
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  • #2
B) ##\binom{3}{1}\binom{37}{4}+\binom{3}{2}\binom{37}{3}+\binom{3}{3} \binom{37}{2} = 222\,111##

C) ##P(E) = \frac{N(E)}{N(S)} = \frac{222\,111}{658\,008} \approx 33.76\%##
 
  • #3
Your second attempt looks good.

Another way to calculate (B) is to figure out how many ways to select 5 good boards and then subtract that number from your answer to (A).
 

1. What is the definition of a combination of samples problem?

A combination of samples problem is a mathematical concept that involves selecting a specific number of items or elements from a larger set without regard to the order in which they are chosen.

2. How is a combination of samples problem different from a permutation problem?

A combination problem involves selecting items without regard to their order, while a permutation problem involves selecting items with a specific order in mind.

3. What is the formula for calculating the number of combinations in a sample problem?

The formula for calculating combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items in the set and r is the number of items being selected.

4. Can you provide an example of a combination of samples problem?

One example of a combination of samples problem is selecting a team of 3 players from a group of 8. In this scenario, the order in which the players are chosen does not matter, so it is a combination problem. The number of possible combinations would be 8C3 = 56.

5. How are combination of samples problems used in real life?

Combination of samples problems are used in various fields such as statistics, genetics, and computer science. In real life, these problems can be used to determine the probability of certain outcomes, analyze data, and make predictions based on the sample data.

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